Course Handout-BMA 101 PDF

Preview

Summary

Course Handout-BMA 101...

Description

Course Handout Course details Faculty name Vikram Kumar Programme B. Tech. Semester I 1 Section 8 Course code BMA101 Course title Multivariable Calculus Vision of SBAS 2 To be recognized globally as a provider of education in Basic and Applied Sciences, fundamental and interdisciplinary research Mission of SBAS  Develop solutions for the challenges in science through value-based science education.  Conduct research leading to innovation in sciences. 3  Nurture students into scientifically competent professionals in the usage of modern tools.  Foster in students, a spirit of inquiry and collaboration to make them ready for careers in teaching, research and corporate world. Programme educational objectives (PEOs) PEO1 4 PEO2 PEO3 PEO4 Programme outcomes PO1 Engineering Knowledge PO2 Problem analysis PO3 Design/development of solutions PO4 Conduct investigations of complex problems PO5 Modern tool usage 5 PO6 The engineer and society PO7 Environment and sustainability PO8 Ethics PO9 Individual or team work PO10 Communication PO11 Project management and finance PO12 Life-long Learning Programme specifics outcome (PSO) (if any) PSO1 6 PSO2 PSO3 7 Course outcomes (COs) CO1 show the convergence of a sequence, series and compute some important series expansions of a single variable function. (K3) CO2 examine mean value theorems for real-valued functions, show the convergence of the improper integral and apply curvature to find evolutes & involutes. (K4)

CO3

CO4 CO5

Evaluation Component Midterm Exam Quiz-1/2/3 8

Online Learning using Swayam/NPTEL /Coursera

10 11 12 13 14 15 16 17

18 19

45 mins

5

Closed Book

Quiz-2

Any time throughout Open Book Assignment(s) the semester Presentation On a (Seminar/mini15 minutes 5 scheduled project/poster) date List of teaching –learning pedagogy: White Board, e- book, Power Point Presentations, Internet Resources, NPTEL Web, Swayam and Video courses. Open hour for students-each faculty will decide Link address for course materialsAssignment(s)

9

use methods to find limit, continuity, derivatives of multivariable scalar functions and relate derivatives to solve the problems of optimization. (K4) apply methods to find integrals of multivariable scalar functions and relate it to solve the problems finding areas and volumes. (K4) explain the three elements of vector differential calculus, apply these elements for evaluation of integrals of vector valued functions and relate the three important theorems to evaluate the problems of integrations. (K5) Duration Marks Date &Time Nature of Evaluation (50) Component Component 90-120 17-24 Oct Closed Book Midterm Exam 50 (30) 2019 mins 15 mins Academic Closed Book Quiz-1 5 calendar each

Within two days

5

Recommended list of e-books. Recommended list of online courses like SWAYYAM/NPTEL/MOOCS etc Recommended list of mini projects / projects/ technical training etc. Students’ Presentation List of e-books List of NPTEL/MOOCS/SWAYAM/Courses/Video 1: Basic Calculus for Engineers, Scientists and Economists(Video). url:www.nptel.ac.in/courses/111104085/ 2: Mathematics-I(web) url: www.http://nptel.ac.in/courses/122101003/ Content beyond Syllabus- each faculty will decide List of mini projects/projects-NA

20

Lec tur e No.

Detail academic calendar of lecture topics

Date

Topics to be covered

  1

2

3

03/9/2019

04/9/2019

05/9/2019

4

06/9/2019

5

11/9/2019

 

               

 

Learning outcomes of each topic

Defini ng:sequence,* conv er genceofa sequence( gr aphi cal l y) Cal cul at i ngl i mi t sofasequence ( Thm 1/576) I l l us t r at i v eexampl es3 Defini t i ons:I nfini t eser i es,sequenceof par t i alsums,andconv er genceofan i nfini t eser i es Geome t r i cser i esandi t sconv er genc e nt ht er mt estf ordi v er genc e.Exa. 7 I nt egr alTes t I l l us t r at i v eexampl e3:Thepser i es Defini t i on absol ut econv er genc e The or em:t heabsol ut econv er gencet es t Ther at i ot est I l l us t r at i v eexampl es2 andt her oott es t I l l us t r at i v eexampl es4 Summar yoft es t s( 614/T2) Powe rser i esandi t sconv er genc e Radi usofconv er genceofapowerser i es I l l us t r at i v eexampl es3

CO1

Defini t i ons:Tayl orandMacl aur i nser i es Expansi onofexponent i al ,t r i gonome t r i c andl ogar i t hmi cser i esandt hei r conv er gence. I l l us t r at i v eexampl es1, 2, 3 Macl aur i nSer i esexpansi onsusi ngknown ser i es

Relate d Unit of syllab us

Total lectur e in the Unit

Unit -1

6

Reference Chap./Sec. (Book) (T1 means test book in serial 1 and so on R1 means reference book in serial no 1 and so on) 10. 1/572/T2 *Donotuse mat hemat i cal

) , r i gour ( usegr aph. 10. 2/584/T2 CO1

10. 2/584/T2

CO1

10. 3/593/T2 10. 5/604/T2

CO1

10. 7/616/T2

CO1

10. 8/626/T2

6

7

12/9/2019

13/9/2019



I l l us t r at i v eexampl es4

 

Four i erhal fr angesi neandcosi neser i es I l l us t r at i v e Exampl es

CO1

9. 3/9. 16/R1



St at ementofRol l e ’ sTheor em andi t s geome t r i cali nt er pr et at i on. * I l l ust r at i v eExampl e8. 1. St at ementofMeanVal ueTheor em andi t s geome t r i calandphysi cali nt er pr e t at i on. I l l us t r at i v eExampl e8. 3.

CO2

2. 8/162/T1

   

8

9

10

11

12

13

17/9/2019



0  , I ndet er mi nat ef or ms0  and ˆ L'Hopital's rule

 

I l l us t r at i v eExampl es6. 1,6. 3 Ot heri nde t er mi nat ef or ms



Exampl es



Defining improper integrals of Type I and their convergence



I l l us t r at i v eexampl es1, 2, 3



Defining improper integrals of Type II and their convergence



I l l us t r at i v eexampl es4, 5

     

Gammaf unct i on:Defini t i on Pr oper t i esofgammaf unc t i on I l l us t r at i v eexampl es Be t af unc t i on:defini t i on Pr oper t i esofbe t af unct i on Rel at i onbet weenbe t aandgamma f unc t i ons Appl i cat i onofbe t aandgammaf unct i ons t oeval uat ei nt egr al s:

19/9/2019

24/9/2019

25/9/2019  

14

26/9/2019

2. 8/164/T1

CO2

CO2

  , 0.,1 , 00 , 0

18/9/2019

20/9/2019

St at ementofTayl or ’ sTheor em andi t s si gni ficance

10. 9/634/T2

 

Defini ngcurv at ur e,ci r cl eofcur vat ur e, r adi usofcur vat ur eandcent r eof cur vat ur e Defini ngEvol ut eandI nvol ut eofacur ve I l l us t r at i v eExampl es1. 67

8. 8/504/T2

CO2 Unit 2

8

CO2

8. 8/508/T2

CO2

1. 5. 4/1. 60/ R1

CO2

1. 5. 4/1. 63/ R1

CO2

1. 6. 3/1. 89/ R1



15

9/10/2019

16

10/10/2019

17

11/10/2019

18

15/10/2019

                   

19

20

16/10/2019



17/10/2019

       

21

18/10/2019

22

22/10/2019

  

23

23/10/2019

  

Defini t i ons:Funct i onsoft woort hr ee CO3 var i abl es,domai n,r ange I l l ust r at i veExampl es1 Leve lcur vesandgr aphsofz =f ( x, y) I l l us t r at i v eExampl es3 * Defini ngl i mi toff ( x, y) . Cal cul at i ngl i mi t susi ngt heor em1 I l l us t r at i v eexampl es1, 2 Pat ht estf ornonexi s t enceofl i mi t s CO3 14. 2/806 I l l us t r at i v eexampl es6/806 Defini ngcont i nui t yoff ( x, y) I l l us t r at i v eexampl es5/805 Defini ngpar t i alder i vat i v esoff ( x, y)i n CO3 t er msofl i mi t s Geome t r i cali nt er pr et at i onofpar t i al der i vat i v es Met hodt oeval uat epar t i alder i vat i v es I l l us t r at i v eexampl es1, 2, 3, 4, 6 Hi gheror derpar t i alder i vat i ves CO3 I l l us t r at i v eexampl es9, 10, 11 Tot alDi ffe r ent i al :defini t i onandr ul est o find I l l us t r at i v eexampl es( seeQ. B. ) CO3 Composi t ef unc t i on Chai nr ul ewhent her ei sonl yone i ndependentvar i abl e Chai nr ul ewhent her ear et wo i ndependentvar i abl es. I mpl i ci tdi ffer ent i at i on I l l us t r at i v eexampl es1, 3, 5 CO3 Defini ngTayl orser i esoff ( x, y) I l l us t r at i v eexampl es(seeQ. B. ) Tayl orser i esoff ( x, y)usi ngknownTayl or ser i esoff ( x) I l l us t r at i v eexampl es( seeQ. B. ) Defini ngl ocalmaxi maandl ocalmi ni ma CO3 Fi r stder i vat i v et estf orl ocalext r eme val ues Cr i t i calpoi nt sandsaddl epoi nt s Secondder i vat i v et es tf orl ocalext r eme val ues CO3 I l l us t r at i v eexampl es3, 4, 5 ( * wewi l lnotdi scussabsol ut emaxi maand mi ni ma. ) Defini ngcons t r ai nedmaxi maandmi ni maCO3 Theme t hodofLagr angemul t i pl i er s I l l us t r at i v eexampl es3, 4, 5

14. 1/793/T2

14. 2/801/T2

14. 2/805/T2

14. 3/810/T2 14. 3/815/T2 14. 6/844/T2 andHandout Unit 3

9

14. 4/821/T2 ( * wewi l lnot di scuss c hangeof var i abl es pr obl ems) 14. 9/866/T2

14. 7/848/T2

14. 7/851/T2

14. 8/857/T2

 24 24/10/2019



 25

25/10/2019

26

30/10/2019

  

27

31/10/2019

  

CO4 Defini ngdoubl ei nt egr al s:( i )ast hel i mi tof afini t esum and,( i i )geomet r i cal l yas vol ume. Met hodofeval uat i ng doubl ei nt egr al sover r ect angul arr egi onsusi ngFubi ni ’ s t heor em

15. 1/882/T2

Met hodofeval uat i ng doubl ei nt egr al sov er CO4 nonr ect angul arr egi onsusi ngFubi ni ’ s t heor em Changeofor derofi nt egr at i on CO4

15. 2/887/T2

15. 2/890/T2 15. 3/896/T2

CO4 I nt r oduci ngpol arcoor di nat es Defini ngdoubl ei nt egr at i oni npol ar coor di nat es Met hodofeval uat i onofdoubl ei nt egr at i on. I l l us t r at i v eexampl es1 ( * i nt r oducet hepol arcur vesCi r cl esr a ,

r a cos ,r a sin car di oi dr  a(1 cos )  28

1/11/2019

5/11/2019

Fi ndi ngar eaandvol umebydoubl e i nt egr at i on Fi ndi ngar easi npol arcoor di nat es

CO4



Defini ngt r i pl ei nt egr al sast hel i mi tofa fini t esum anddeduci ngf or mul af or vol ume Met hodofeval uat i on



Mor epr obl emsoffindi ngTr i pl ei nt egr al s



Fi ndi ngvol umeusi ngt r i pl ei nt egr al s

   

Defini ngscal arfiel dsandv ect orfiel dsCO5 For mul af orfindi nggr adi ent Geome t r i calmeani ngasnor malvec t or Physi calmeani ngasdi r ect i onofgr eat est



 30

31

32

33

6/11/2019

15. 4/900/T2

CO4

CO4

 29

Lemni scat es) Changi ngCar t esi ani nt egr al si nt opol ar i nt egr al s

Unit 4

9 15. 4/902/T2

14. 5/938/T1

1/909/T2

CO4

7/11/2019 CO4

8/11/2019

19/11/2019

15. 3/15. 10/ R1 Unit 5

10

  34

20/11/2019

   

35

21/11/2019 

36

22/11/2019

37

26/11/2019

 

27/11/2019

   

38

 

 39

28/11/2019   

40 41 42

29/11/2019     

r at eofc hange I l l us t r at i v eexampl es15. 11, 15. 13, 15. 16 For mul ast ofinddi v er genceandcur lof v ect orfiel ds Pr oper t i esofdi v er genc eandcur l I r r ot at i onalfiel dsandI nc ompr essi bl e fie l ds I l l us t r at i v eexampl es15. 19, 15. 20 Physi cali nt er pr et at i onofdi v er genc eand cur l

CO5

15. 4/15. 18/ R1

CO5

15. 21/R1

CO5 Defini ngLi nei nt egr al sast hel i mi tofa fini t esum andwor kdonebyaf or cefie l d Met hodst oeval uat eLi nei nt egr al s I l l us t r at i v eexampl es15. 27,15. 28, 15. 31

15. 5/15. 25/ R1

CO5 Pat hi ndependence Conser vat i vefiel dandpot ent i alf unc t i on CO5 St at ementofGr een’ sTheor em Appl i cat i ont oeval uat eLi nei nt egr al s Eval uat i ngpl anear ea I l l us t r at i v eexampl es15. 35

15. 5. 1/15. 30 /R1 15. 5. 2/15. 34 /R1

Defini ngsur f acei nt egr alofavec t orfie l d CO5 ast hel i mi tofafini t esum andasfluxofa v ect orfiel d. Met hodt oeval uat e I l l us t r at i v eExampl es15. 45 CO5 Sur f acei nt egr al sov erCyl i ndr i caland s pher i calsur f aces I l l us t r at i v eexampl es( seeQ. B. ) St at ementofSt okes’ st heor em CO5 Appl i cat i ont ofindl i nei nt egr al s CO5 St at ementofGauss’ st heor em Appl i cat i ont ofindsur f acei nt egr alov er Cubi calands pher i calsur f acesonl y

15. 6. 2/15. 53 /R1

Course Description: Even though the first two units deal with some of the concepts of the single variable calculus that are prerequisites for the other engineering mathematics courses, the core of the course is the multivariable calculus.

15. 7. 2/15. 63 /R1 15. 7. 1/15. 56 /R1

Unit-I deals with convergence of series followed by some special series expansions of functions of a single variable and their convergence. Unit-II begin with learning some of the fundamental theorems of differential calculus and their utility in constructing a rule using which we can evaluate limits of indeterminate forms. We also study improper integrals and two special functions (improper integrals), namely, beta and gamma functions. Unit-III deals with differential calculus of multivariable functions. It begins with defining a function of two variables, its limit and continuity followed by defining its derivative (partial derivative) and applications of partial derivatives in the form of the problem of finding Maxima and Minima. Unit-IV deals with methods to evaluate double and triple integrals and their applications in finding Plane areas and volumes of solids. We also introduce 2d polar coordinates and its changing in Cartesian coordinates. Unit-V gives the idea about the calculus of vector functions. In the differential calculus part we study three basic elements of vector calculus, namely, Gradient, Divergence and Curl and their properties. In the integral calculus part we first learn the techniques of evaluating line integrals and surface integrals followed by the three all important theorems, namely Green’s, Stokes’s and Gauss Divergence theorem using which we can easily evaluate line and surface integrals Text Books: T1. Robert T. Smith and Roland B. Minton, Calculus, 4th Edition, McGraw Hill Education. T2. George B. Thomas and Ross L. Finney, Calculus and Analytic Geometry, 9th Edition, Addison Wesley Publishing company, 1995. Reference Books: R1. R. K. Jain and S. R. K. Iyengar, Advanced Engineering Mathematics, 4th Edition, Narosa publishers. R2. Michael D. Greenberg, Advanced Engineering Mathematics, 2nd Edition, Pearson Education

Course Content

Module-I Contact Hours: 6 Convergence of sequence and series, tests for convergence; Power series, Taylor's series, series for exponential, trigonometric and logarithm functions; Half range Fourier sine and Fourier cosine series. Module-II Contact Hours: 8 Evolutes and involutes, Rolle’s Theorem, Mean value theorems, Taylor’s and ˆ ' s rule , Maclaurin theorem with remainders; indeterminate forms and L ' Hopital Evaluation of definite and improper integrals; Beta and Gamma functions and their properties. Module-III Contact Hours: 9 Functions of several variables, Limits and continuity, Partial derivatives, Total differential, Derivatives of composite and implicit functions, Extreme values and saddle points, Lagrange’s method of undetermined multipliers. Module-IV Contact Hours: 9 Double integrals in Cartesian and Polar coordinates, Change of order of integration, change of variables (Cartesian to polar), Applications of double integrals to find area and volume, Triple integrals in Cartesian, Aapplications of triple integral to find volume. Module-V Contact Hours: 10 Scalar and vector fields, Differentiation of Vector functions, Gradient, divergence, curl, line integrals, path independence, potential functions and conservative fields, surface integrals, Green’s theorem, Stokes’s theorem and Gauss’s divergence theorem (without proof & simple problems).

Life-long

Project

Communicati

Individual or

Ethics

Environment

The engineer

Modern tool

Conduct investigations

Design/devel

Problem

Engineering

Program Outcomes(POs):

PO 8

Learning

PO 7

management and finance

PO 6

on

PO 5

team work

of complex problems PO4

and sustainability

opment of solutions PO 3

and society

analysis PO 2

usage

Knowledge PO 1

PO 9

PO1 0

PO1 2

PO11

Course outcomes (COs) and Program Outcome (POs)Mapping: CO/PO Mapping (S/M/W indicates strength of correlation) S-Strong, M-Medium, L-Low COs Programme Outcomes(POs) PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 CO1 CO2 CO3 CO4 CO5

3 3 3 3 3

2 2 2 2 2

2 2 2 2 2

2 2 2 2 2

1 1 1 1 1

PO11

1 1 1 1 1

PO12 1 1 1 1 1

Compliance report School of Basic and Applied Science Programme Programme Chair Sl No

Course code

Sigature of PC;

Compliance report of course handout Taught Course Course Sectio Course handout by coordinato Submission date title n faculty r

Signature of Dean:

Remark s by PC if any...